Though your idea sounds interesting and is plausible by the rules, tactically, I would highly suggest against it, as it causes more problems than it solves.

You're creating a beacon to attract enemies.

The wording of the spell states that, though you are invisible to everyone outside of the extra-dimensional space, the only way to truly hide would be to pull the rope into the space with you. However, revising the PHB, both pulling the rope up and dropping it would fall under "Using an Object" and would cost you an action each. Even if the DM is nice and considers it a bonus action, you'll still be unable to do both within one move.

That would mean that you'll be leaving behind a 7-foot tall pointer leading straight to you while alerting your enemies that you're a high-priority target. This will surely result in attracting many melee enemies to the base of your rope as it will create a focal point for ranged attackers.

Note: This problem is also seen when players try to use illusionary spells like Mordenkainen's Private Sanctum to avoid being targeted, but then jump in and out of it all the time... the enemy still knows where you are.

Enemies can strategize too.

Having the rope exposed might not seem detrimental, but remember that your enemies have a certain level of intelligence, and won't take long before choosing to encumber you. This opens up three Opportunity Attacks that your character will face:

1. Ready is an action available to enemies as well as PCs, meaning you can face up to 24 melee attackers waiting for you to climb down (8 opponents with regular weapons and 16 with reach weapons) If each opponent chooses to use Ready, then the minute your feet hit the floor, you got a massacre of priority attacks coming your way.

2. Ranged Ready. The same as before, but with ranged attacks, they can choose to hit you when you're fully outside of the space, but still at the top of the rope. Failing a saving throw here would both cause falling damage and most likely prone as you need to refocus to get up.

3. Even without ranged attacks, an enemy with enough reach can cut the rope short. Since the spell says nothing about the spell being broken if the rope is cut, all they would have to do is set a trap for when you fall, cut the rope and wait up to an hour. You'll either try to get out or the spell will cancel and you'll fall right into a pit of swords, a bear trap, or anything else they can come up with. Meanwhile, the bulk of the enemy can focus on wiping out your party, either slaughtering them or causing them to think you a coward for your cunning plan.

4. All it takes are two melee fighters at the base of your rope and climbing would also cause Opportunity Attacks every time you leave.

All in all, to cast one spell will cost you many prioritized hits.

You cannot use the space as half-cover

Reading the RAW, the way to leave the extra-dimensional space is to climb down. This means that unless you have enough room to utilize the rope, you'll be stuck in there for an hour. I could be wrong, but since climb is the key word there, it defeats the purpose of the sanctum if you don't use the rope as a failsafe passage. Though, if you found a way to creatively word your entering and exiting as climbing, you're still faced with being a priority.

All in all, the one key problem I can see in your plan is in how enemies prioritize their targets:

Swords are troublesome. Flying things are more troublesome. Tricky things are most troublesome. Must work on tricky enemy first.

The rules allow this, but...

There's nothing in the rules that prevents this from happening. However, there are a few things to consider about this tactic before employing it.

It can be immersion-breaking

Although it is technically within the rules of DnD to abuse the turn-based combat system for janky things like this, for some players this can break the sense of immersion that comes with good DnD. It can transform the fight from an epic tale about spell-slinging wizards to a videogame where you abuse the mechanics for maximum power. This kind of tactic is legal in the game, but does not make for a good story or in-universe character.

Any good DM can (and should) respond appropriately to shenanigans like this

For example, were I the DM here's how it'd play out. The first time you use Rope Trick as a sanctuary, some enemies might try to follow you up but it would mostly work out - this is what the spell is for, and it's a fun part of the game.

However, once you try the "climbing up and down" shenanigans, I'd just have enemies start readying actions. If they ready an attack, they can just attack you while you're on the rope. You could also have enemies stand at the bottom of the rope and attack your allies, and use attacks of opportunity whenever you attempt to climb back up. I also think it would be very reasonable to grant them advantage while attacking a climbing target. At this point, climbing up and down each turn is actually helping your enemies.

At the end of the day, there are a whole host of ways that climbing in and out is a poor tactical decision. That tactic leaves you spending a lot of time climbing up and down a rope, which is a poor use of movement and limits your positioning to a single location. While climbing you are fully exposed and can't use hands, but your enemies are free to reposition at will during their turns. Any slightly-intelligent enemy could use this highly advantageous situation to make your life miserable in a variety of ways.

You're missing the best use of Rope Trick

Rope trick is a fantastic spell for the use case you mentioned in the question:

[to] give a character in dire condition a brief respite against ranged attacks or spells.

However, it works better thematically and mechanically to stay in the sanctuary. You can take a few turns to use healing potions and spells, regroup, and reconsider tactics. You could even have your whole party take a short rest. Popping in and out every turn can be immersion-breaking and actually hurt more than it helps mechanically.